HNN extensions of inverse semigroups
Tatiana B. Jajcayová · Lincoln (University of Nebraska) · 1997
The concept of HNN-extensions of groups was introduced by Higman, Neumann and Neumann in their study of embeddability of groups in 1949. In combinatorial group theory, HNN-extensions play an important role in applications to algorithmic problems. Recent work of Yamamura introduces HNN-extensions in the category of inverse semigroups. His work shows the usefulness of HNN-extensions in this class by proving the undecidability of any Markov property as well as the undecidability of several non-Markov properties for finitely presented inverse semigroups. In our work we study the structure of HNN-extensions of inverse semigroups by investigating properties of their Schutzenberger graphs. We show that every such graph has a tree-like structure. This fact is used to describe the subsemigroup generated by the stable letter t in an HNN-extension. We introduce a certain class of HNN-extensions of inverse semigroups that we call lower bounded HNN-extensions for HNN-extensions in this class we provide an iterative procedure for building Schutzenberger automata. In certain cases this procedure yields an effective construction of the Schutzenberger automata and thus provides a solution for the word problem. We analyze conditions under which the procedure is effective and show that the word problem is solvable in particular for HNN-extensions of free inverse semigroups. Further, we characterize Schutzenberger automata corresponding to lower bounded HNN-extensions. Such automata are characterized as automata that have a certain lobe structure and that contain a special subgraph with certain finiteness properties. This characterization is then used along with techniques from the Bass-Serre theory of graphs of groups to give a description of the maximal subgroups contained in a lower bounded HNN-extension. We show that the class of lower bounded HNN-extensions includes many well known inverse semigroups.